解题方法
1 . 已知函数
.
(1)用单调性定义证明:
在
上单调递增;
(2)若函数
有3个零点
,满足
,且
.
①求证:
;
②求
的值(
表示不超过
的最大整数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4247d7790d83be16bc74aa5e5d12dd63.png)
(1)用单调性定义证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed6d804ef44bfc64f824b0ccef71765e.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f8994d83bf4a688c0ab897a5a40fa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a1cc5cfec94bc5686b41b043acdc8ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1310a7a80d1f8751a3f8cafe7f8c8b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d995c5d2e1e0305d805032e18997986a.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f28cbe8f17c4472d8663f9ccbe3b98f6.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59077d1948911b13d68a572eadbca3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
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解题方法
2 . 在数学中,不给出具体解析式,只给出函数满足的特殊条件或特征的函数称为“抽象函数”.我们需要研究抽象函数的定义域、单调性、奇偶性等性质.对于抽象函数
,当
时,
,且满足:
,均有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
在
上单调递增;
(2)若函数
满足上述函数的特征,求实数
的取值范围;
(3)若
,求证:对任意
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c73a98c1b3504e09bfbe0db849b0d24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6571b33b56c6cd88f2f6e091031bcf40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/367936b458618efb6b2eadc843e5d6ba.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d562dc22dfb3b81d0c3f88b54d063c2f.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/639c5f8b7a1a268c904d04356f0d1b3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/249a976e88133f3b3733f09137cf5c42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3be9b79f42bbf0de1851607050c3e8d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/219598f1289ddb370d632ea141731d52.png)
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3 . 对于项数为
的数列
,若数列
满足
,
,其中,
表示数集
中最大的数,则称数列
是
的
数列.
(1)若各项均为正整数的数列
的
数列是
,写出所有的数列
;
(2)证明:若数列
中存在
使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce3b6654490dcd8177970631e929d3d.png)
,则存在
使得
成立;
(3)数列
是
的
数列,数列
是
的
数列,定义
其中
.求证:
为单调递增数列的充要条件是
为单调递增数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfd88e35dc6c2b82b4bb29475d37c015.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cda05b106f920bbcfa02320229ca3c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698f45c9ed5bb04924f1037107e76988.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)若各项均为正整数的数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c2f32a17f0261d32079efed31d414a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
(2)证明:若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cce3b6654490dcd8177970631e929d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6afadbd5dc4d8003ac2a0c85678dbecf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4578089af6806bf1257491091b924d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c2dd6a827492dffddd07e621a4bbe36.png)
(3)数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c3bd239656c2ef509ed9f2a91a68317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb951647594b67c82045b7ba69f57cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4af75cc7d8cc976dce8bf9bd8fdc18f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa9f328d5108bbec5c56eebfe95567.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d019d9549df6aed0dab378301d889ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70322214e5c9d9a8df10eb45930f5745.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb2e86a5aa896bef041701e0e1771ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8504784fec514a92d845910c6721c3a.png)
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解题方法
4 . 若函数
在定义域内存在两个不同的数
,
,同时满足
,且
在点
,
处的切线斜率相同,则称
为“切合函数”.
(1)证明:
为“切合函数”;
(2)若
为“切合函数”(其中
为自然对数的底数),并设满足条件的两个数为
,
.
(ⅰ)求证:
;
(ⅱ)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf7c745cd02f4620a175cf00ec85e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a24a2c53e3b0b1c08803e95419f909d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecaca8409b3f51d22667a14559c58ea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d275fbb3ee5cd1177ca5a2ceecbbef0f.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbe0de54dfc96a2291e8d5e56676eabc.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fb46178ba0560d96bd3a05891505b95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
(ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c20b8bd265b07dd90690ad4e349c6dc.png)
(ⅱ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84cde09c609543feedc2e0c11992b2bd.png)
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2024-01-03更新
|
1043次组卷
|
4卷引用:重庆市南开中学校2024届高三上学期第五次质量检测数学试题
重庆市南开中学校2024届高三上学期第五次质量检测数学试题重庆市沙坪坝区南开中学校2024届高三上学期第五次质量检测数学试题江西省赣州市南康中学2024届高三上学期新高考“七省联考”考前数学猜题卷(一)(已下线)微考点2-5 新高考新试卷结构19题压轴题新定义导数试题分类汇编
5 . 《几何原本》是古希腊数学家欧几里得得所著的一部数学著作,在《几何原本》第六卷给出了内角平分线定理,其内容为:在一个三角形中,三角形一个内角的角平分线内分对边所成的两条线段,与这个角的两邻边对应成比例.例如,在
中(图1),
为
的平分线,则有
.
(2)如图2,已知
的重心为
,内心为
,若
的连线
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/608bf0cfbbe809837adec2755fcd2901.png)
(2)如图2,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e105760638b22b26ff8bec4354255e4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57b8fc74eea80b1ccf11d16ad7b3178a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/981b01ddc1aa5fcf155ad41307d22b17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a94a70686cb9c91ec9705bed47dc663.png)
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名校
6 . n个有次序的实数
,
,
,
所组成的有序数组
称为一个n维向量,其中
称为该向量的第
个分量.特别地,对一个n维向量
,若
,
,称
为n维信号向量.设
,
,
则
和
的内积定义为
,且
.
(1)直接写出4个两两垂直的4维信号向量.
(2)证明:不存在14个两两垂直的14维信号向量.
(3)已知k个两两垂直的2024维信号向量
,
,
,
满足它们的前m个分量都是相同的,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efee470d0232b6b37f2fb2ab15aae0ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d5cd21ff3c760e7ec3130f5bfa8c91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da796531c7b6c590a22b811df1fcef53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0cf905f1d4af294ebc9c19facd64c3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32d5cd21ff3c760e7ec3130f5bfa8c91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a414d372b680499f1c8ca1a7ae5f4d82.png)
则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64c5562bd4d1b54424330cb6329cd79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45ba716f03748c19b7ce2f99af536ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd04a2f39ff6f36e9531bac16960d71e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19c7c807358869b70becd16ca80e1714.png)
(1)直接写出4个两两垂直的4维信号向量.
(2)证明:不存在14个两两垂直的14维信号向量.
(3)已知k个两两垂直的2024维信号向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/596afe6f8149e39c53d36a759bee6151.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf2182d0dad848ccc76944d976befbf2.png)
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7 . 在平面直角坐标系xoy中,已知
,圆C:
与x轴交于O ,B.
(1)证明:在x轴上存在异于点A的定点
,使得对于圆C上任一点P,都有
为定值;
(2)点M为圆C上位于x轴上方的任一点,过(1)中的点
作垂直于x轴的直线l,直线OM与l交于点N,直线AN与直线MB交于点R,求证:点R在椭圆上运动.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6316e0e6da742e9b035d8f2cc91a4dd.png)
(1)证明:在x轴上存在异于点A的定点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/439e95540157803d4ac3cf61a49f50a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c7383714dc2ac9fe164e26a4d1bbd0c.png)
(2)点M为圆C上位于x轴上方的任一点,过(1)中的点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/439e95540157803d4ac3cf61a49f50a8.png)
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8 . ![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
存在唯一的零点
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
的零点记为
,设
,求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b57ed792fb63c756aa4372e501f73cf.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/270ce4af9a3c5f7987ddef4988ae0a57.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b752f0f189e5d8666daea73e145dff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e5ad7a134838f6ee246e606a625f7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eb3c14b2ab08a915682646f3377b7b4.png)
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2023-10-01更新
|
159次组卷
|
3卷引用:专题04 指数函数与对数函数2-2024年高一数学寒假作业单元合订本
(已下线)专题04 指数函数与对数函数2-2024年高一数学寒假作业单元合订本福建省漳州实验高级中学2022-2023学年高一创新班上学期期中考试数学试题福建省厦门市厦门二中2023-2024学年高一上学期12月月考数学试题
9 . 在平面直角坐标系
中,P,Q是抛物线
上两点(异于点O),过点P且与C相切的直线l交x轴于点M,且直线
与l的斜率乘积为
.
(1)求证:直线
过定点,并求此定点D的坐标;
(2)过M作l的垂线交椭圆
于A,B两点,过D作l的平行线交直线
于H,记
的面积为S,
的面积为T.
①当
取最大值时,求点P的纵坐标;
②证明:存在定点G,使
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8eb9603390e28948abd2e3cd96e1720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274a9dc37509f01c2606fb3086a46f4f.png)
(1)求证:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(2)过M作l的垂线交椭圆
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c82e7d9f4f7ace849e09e9adcb786b7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea1f0417d8269f01d8e0bc1a8756e2ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895f5fcb1cebfc09dc14ab6efad03437.png)
②证明:存在定点G,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac90c4158636c076ef1d0d45df68be88.png)
您最近一年使用:0次
2023-05-08更新
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942次组卷
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5卷引用:通关练17 抛物线8考点精练(3)
(已下线)通关练17 抛物线8考点精练(3)山东省烟台市2023届高考适应性练习(一)数学试题山东省枣庄市2023届高三三模数学试题(已下线)高二上学期期中复习【第三章 圆锥曲线的方程】十二大题型归纳(拔尖篇)-2023-2024学年高二数学举一反三系列(人教A版2019选择性必修第一册)湖南省株洲市第二中学2022届高三下学期第三次月考数学试题
10 . 已知
,
(1)求函数
的导数,并证明:函数
在
上是严格减函数(常数
为自然对数的底);
(2)根据(1),判断并证明
与
的大小关系,并请推广至一般的结论(无须证明);
(3)已知
、
是正整数,
,
,求证:
是满足条件的唯一一组值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f53f81bca037a4383c1fab122a3cd3d.png)
(1)求函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17b4888d8cf85f200763db925ce501b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/041a7c8fc017f596542c5e6ec7d1c40b.png)
(2)根据(1),判断并证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8520e118f7e2aab0cea0fc23c833ccbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f15d2a3cd491be27bc3d8799b3f9f610.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a46e678bf9d2df5ad4c782b3dc22f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efdc0e0ca559f0f1af6127545f356fa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20e1c681b27df538bd4742f6cd8298ae.png)
您最近一年使用:0次
2022-12-15更新
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809次组卷
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6卷引用:上海市高二下学期期末真题必刷01(易错题)--高二期末考点大串讲(沪教版2020选修)
(已下线)上海市高二下学期期末真题必刷01(易错题)--高二期末考点大串讲(沪教版2020选修)(已下线)上海市高二下学期期末真题必刷04(压轴题)--高二期末考点大串讲(沪教版2020选修)上海市嘉定区2023届高三上学期一模数学试题重庆市2023届高三下学期2月月度质量检测数学试题(已下线)核心考点09导数的应用(1)上海市静安区市北中学2024届高三上学期12月月考数学试题